one obtains the classical trajectories \({\bf q}^\ast (t), {\bf p}^\ast (t)\)  from
these Hamiltotian equations (REF ). We have also to specify the mean field in
the Hamilton function,
\(H_N({\bf q},{\bf p})=\sum _\kappa \left[{\bf p}^2_\kappa /2m + V_\kappa ({\bf q}_\kappa )\right])\) .
If we are
far from the bifurcations
(CPs), one can use the standard SPM, and the non-zero 2nd order terms
of the entropy expansion.
Such derivations lead to
the results of the standard thermodynamics but near the CP.
Near the
bifurcation, where the second order terms of the entropy expansion is zero, we
may employ the improved SPM (ISPM) [1]}, [2]}
transforming the ISPM for the action
phase integral of the POT [3]}, [3]}, [5]}, [6]}, [2]} to the real
exponent argument – the entropy \(S\) ,
Eq. (REF ). The simplest ISPM is the second - order expansion of the
entropy (REF ), but with finite integration limits.
Now, one can take the path integral
analytically
in terms of the \(\mbox{erf}\)  functions of the real argument.
Here is the place where we can apply for the catastrophe theory of Fedoryuk
[8]}, [9]} by expanding the entropy
to the third order terms. In this way, we arrive at the Airy-kind integrals
with the finite contributions of the two SPM points which turn into one
bifurcation point at the limit to the CP.
Note that in order to remove singularity of the fluctuations \(\omega \)  near at the critical point
with generalization to the QS description, one can calculate \(\omega \) 
through the moments of the statistical level density \(\rho (E,A) \)  [10]}, also with
using the ISPM.
